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Homework Help: Laser Operation - Gain Media

  1. Nov 29, 2013 #1
    I'm not sure if this question should be posted in the introductory physics section or the advanced physics section.

    Consider an amplifying medium, composed of homogeneous broadening four-level atoms as show in figure 26.5, page 557 of textbook.

    http://img689.imageshack.us/img689/6108/lt2f.png [Broken]

    Amplification is to occur on the 2-to-1 transition. The medium is pumped by a laser of intensity [itex]I_{p}[/itex], which is resonant with the 3-to-0 transition. The spontaneous decay processes are indicated on the diagram. The total number of gain atoms is [itex]N_{T} = N_{0} + N_{1} + N_{2} + N_{3}[/itex]. The various parameters are:

    [itex]k_{32} = 10^{8} \frac{1}{s}; k_{21} = 10^{3} \frac{1}{s}; k_{10} = 10^{8} \frac{1}{s}; k_{30} = k_{31} = k_{20} = 0[/itex]
    [itex]σ_{p} = 4x10^{-19} cm^{2}; σ = 2.5x10^{-18} cm^{2}; λ_{30} = 300 nm; λ_{21} = 600 nm; N_{τ} = 2.0x10^{26} m^{-3}[/itex]

    Assuming an ideal four-level laser system determine:

    a) The pump irradiance required to sustain a small signal gain coefficient of [itex]\frac{0.01}{cm}[/itex]
    b) The saturation innradiance.

    2. Relevant equations
    The small-signal gain coefficient [itex]γ_{0}[/itex] and the saturation irradiance [itex]I_{S}[/itex] take the form

    [itex]γ_{0} = σR_{p2}\tau_{2}[/itex]

    [itex]I_{S} = \frac{hv^{'}}{σ\tau_{2}}[/itex]
    [itex]R_{p2}[/itex] is a effective pump rate density

    [itex]R_{p2} = \frac{κ_{32}}{κ_{3}}(\frac{σ_{p}I_{p}}{hv_{p}}N_{T})[/itex]
    In a closed system

    [itex]κ_{3} = κ_{32} + κ_{31} + κ_{30}[/itex]
    The lifetime [itex]\tau[/itex] of an energy level is defined to be the inverse of the total decay rate from the level so that

    [itex]\tau_{n} = \frac{1}{κ_{n}}, n \epsilon Z[/itex]
    Planck's constant [itex]h\ =\ 6.62606876(52)\ \times\ 10^{-34}\ J\ s[/itex]
    The phase velocity [itex]v_{p}[/itex] of a wave can be expressed as

    [itex]v_{p} = \frac{ω_{p}}{k_{p}} ≈ \frac{ω}{k}[/itex]
    [itex]k[/itex] is the propagation constant of a wave that can be expressed as

    [itex]k = \frac{2\pi}{λ}[/itex]

    Where [itex]λ[/itex] is the wavelength
    The angular frequency [itex]ω[/itex] of a wave can be expressed as

    [itex]ω = 2\pi f[/itex]

    Where [itex]f[/itex] is the frequency
    [itex]\pi ≈ 3.14[/itex]

    3. The attempt at a solution

    I start off with the equation for the small-signal gain coefficient [itex]γ_{0}[/itex]

    [itex]γ_{0} = σR_{p2}\tau_{2}[/itex] [1]

    and plug in the equation for [itex]R_{p2}[/itex] effective pump rate density

    [itex]R_{p2} = \frac{κ_{32}}{κ_{3}}(\frac{σ_{p}I_{p}}{hv_{p}}N_{T})[/itex] [2]

    into [1].

    This yields

    [itex]γ_{0} = σ\frac{κ_{32}}{κ_{3}}(\frac{σ_{p}I_{p}}{hv_{p}}N_{T})\tau_{2}[/itex]

    I solve this equation for the pump irradiance [itex]I_{p}[/itex] and get

    [itex]I_{p} = \frac{γ_{0}κ_{3}hv_{p}κ_{2}}{σκ_{32}σ_{p}N_{T}}[/itex] [3]

    I know that for a closed system

    [itex]κ_{3} = κ_{32} + κ_{31} + κ_{30}[/itex]

    Looking at the given variables I get

    [itex]κ_{3} = κ_{32} + 0 + 0 = κ_{32}[/itex]

    Substituting this into [3] yields

    [itex]I_{p} = \frac{γ_{0}κ_{32}hv_{p}κ_{2}}{σκ_{32}σ_{p}N_{T}}[/itex] [3]

    Simplifying this yields

    [itex]I_{p} = \frac{γ_{0}hv_{p}κ_{2}}{σσ_{p}N_{T}}[/itex] [4]

    At this point it looks like I'm very close to solving this problem as all but one variable the phase velocity [itex]v_{p}[/itex] is given. As mentioned in the relevant equations

    [itex]v_{p} = \frac{ω}{k}[/itex]

    This however doesn't really help me. So there must be some other way of expressing the phase velocity [itex]v_{p}[/itex] that I'm not aware of. Once I figure this out I should be able to solve this problem easily. My book doesn't have any examples in this section and I can't seem to find similar questions on the internet, hence I'm stuck and not really sure how to proceed.

    Thanks for any help.
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Nov 29, 2013 #2
    I have just read the caption of the figure and now realize that [itex]v_{p}≈\frac{E_{3} - E_{0}}{h}[/itex]. The only problem now is that I don't know [itex]E_{3}[/itex] or [itex]E_{0}[/itex]. Looks like I might be able to solve this.

    I was able to solve the problem. My only concern is that I didn't use the given wavelengths in the problem. Are they needed to solve the problem?
    Last edited: Nov 29, 2013
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